3.0 PROBABILITY, RANDOM VARIABLES AND RANDOM PROCESSES

Transcription

1 3.0 PROBABILITY, RANDOM VARIABLES AND RANDOM PROCESSES 3.1 Introduction In this chapter we will review the concepts of probabilit, rom variables rom processes. We begin b reviewing some of the definitions of probabilit. We then define rom variables densit functions, review some of the operations on rom variables. We conclude b defining rom processes discussing some properties of rom processes that we will need in our Kalman filter formulations. 3. Probabilit 3..1 Background Suppose we roll a fair die. We sa that the probabilit of a one appearing (the face of the die with one dot) is one-sixth. We arrived at this conclusion b considering that the die has six sides that there is an equal chance that an one side will be facing up when the die ls. This approach to determining probabilit is termed the relative frequenc approach is suitable for defining probabilities associated with simple experiments. With this approach, we determine probabilities from the underling experiment or process deductive reasoning. For example, in the die example, we know that if we roll a die that one of six faces will show. We also assume that it is equall probable that an of the six faces will show. Given this information we thus deduce that the probabilit of an one face showing is one-sixth. For more complex experiments, where the relative frequenc approach becomes more difficult to appl, we can use the axiomatic approach to determine probabilities. The axiomatic approach is based on three probabilit axioms uses them, set theor, to determine probabilities associated with certain experiment outcomes. There is a third approach to determining probabilities that is termed the experimental approach. In this approach probabilities are determined b conducting man experiment tries analzing the results. An example of the experimental approach is the method b which automotive fatalit statistics are derived. To proceed further we must define some terms. We will use the die example to do so. In the die example, the rolling of the die is termed an experiment the appearance of a particular face is an outcome of the experiment. Thus, the die rolling experiment consists of six possible outcomes. If the experiment were the flipping of a coin there would be two outcomes: heads tails. If we were to consider an experiment where we rolled two dice we could define either 36 outcomes or 11 outcomes, depending upon the how we decided to phrase the problem, what probabilities we were tring to find. For the first case the outcomes would be the face with one dot on die one the face with one dot on die two, the face with one dot on die one the face with two dots on die two, so forth. For the second case the outcomes would be the sum of the dots on the two dice equals two, the sum of the dots on the two dice equals three, so forth. M. Budge, Jr Jan

2 3.. Set Theor the Axioms of Probabilit When we discuss probabilities we talk about probabilities associated with events. An event is a collection of outcomes. Thus for the die problem, example events could be: the face with one dot or the face with one dot or the face with two dots. In the first case, the event consists of one outcome in the second case the event consists of two outcomes. This idea of using events to represent collections of outcomes leads to the use of set theor since set theor gives us a means of collecting objects. The analog here is that events are sets outcomes are set elements. There are two events that pla special roles in probabilit theor. These are the certain event the null event. The certain event contains all possible experiment outcomes is analogous to the universe in set theor. It is denoted b. The null event contains no outcomes is analogous to the empt set. It is denoted b. In summar, we will use set theor to represent events we associate probabilities with events. Thus, for the die experiment we can define an event as face with one dot or face with three dots or face with five dots f1,f3,f5 The probabilit of the event occurring is. (3-1) 1 P. (3-) We note that we could use set theor to write f1 f 3 f5 (3-3) which would allow us to write P P. (3-4) This is where the axiomatic approach to probabilit becomes useful. It allows us to use set theor derivations to compute complex probabilities from simple probabilities. as The axioms of probabilit are as follows 1. P 1,. P 0 3. If are mutuall exclusive then P P P. Recall that two sets (events) are mutuall exclusive if. Stated another wa, two events are mutuall exclusive if the appearance of one M. Budge, Jr Jan

3 precludes the appearance of another. Thus if f1 f then are mutuall exclusive because if the roll of the die results in the face with one dot showing, the face with two dots can t be showing, or is precluded from occurring. As indicated earlier, the use of the above axioms set theor allows us to derive man other probabilit relations. Two of the more important ones are P 0 (3-5) P P P P. (3-6) Another important probabilit relation concerns the intersection of two events. This relation is usuall stated as a definition since it cannot be derived from set theor the axioms of probabilit. The probabilit relation states that if two events are independent then P P P. (3-7) We sa that two events are independent if the occurrence of one event doesn t give an information about, or influence the occurrence of, the other event. For example, if we consider the experiment of rolling a pair of dice, the event consisting of a one for die one is independent of the event consisting of a three for die two. This is because we assume that what we roll on the first die will not influence what we roll on the second die. It should be noted that two events can t be mutuall exclusive independent, unless one of the events is the null event. For those who are interested in the finer points of probabilit theor independent events, it should be noted that one can t reall discuss independent events on a single experiment. The can onl be defined on combined experiments. The reader is referred to Papoulis (Papoulis 1991) or some other text on probabilit theor for further discussions of this area. 3.3 Rom Variables Background In the discussions of the previous section we referred to outcomes of the die experiment as face with one dot, etc. If we were discussing the die experiment less formall, we would refer to the rolling of a one. What we have done is assign numbers to the experiment outcomes. When we assign numbers to experiment outcomes we define a rom variable for the experiment. Aside from notational convenience, there is another reason for defining rom variables for experiments: our mathematics background. Specificall, we have developed a wealth of mathematical tools for manipulating numbers. We have also devoted a great deal of time to relating phsical processes to numbers. Unfortunatel, we don t have a wealth of mathematical tools for sets we onl have limited abilit to relate sets elements to phsical processes. For this reason, we use rom variables to take probabilit theor out of the M. Budge, Jr Jan

4 context of set theor put it in the context of mathematics that we know how to work with Definition of a Rom Variable We formall define a rom variable as follows: For ever experimental outcome,, we assign a number. With this we have defined a function x x, which is a mapping from a set of s to a set of numbers. This function is termed a rom variable. In all cases that are of concern to us, we define x as the numerical value of the outcome, x. An example of, i.e., this is the die problem we just discussed. Another example is the voltage across a resistor or the range to a target. It should be noted that we are not required to use such a convention (Papoulis 1991). However, it is convenient. In the future we will not explicitl indicate the dependenc of the rom variable on the experiment outcome. That is, we will replace x b x. We do this for two reasons. One is convenience the other is to force us to stop explicitl thinking about sets think in terms of numbers. This is somewhat subtle but is essential to preventing confusion when studing rom variable theor. There are two basic tpes of rom variables: discrete continuous. Discrete rom variables can have onl a finite number of values continuous rom variables can take on a continuum of values (i.e., there are an infinite number of them). An example of a discrete rom variable is the rom variable associated with the die experiment. An example of a continuous rom variable is the voltage across a resistor Distribution Densit Functions We define the probabilit x x P x F x (3-8) as the distribution function of the rom variable x. In the above x x x x is the event consisting of outcomes,, such that the rom variable associated with those outcomes, x, is less than, or equal to, the number x. The distribution function of the rom variable x is the probabilit associated with this event. This is a long-winded word description that we don t use. In its place we sa that the distribution function of x is the probabilit that x is less than, or equal to, x. When we make the latter statement we should realize that it is a shorth notation for the longer, more formal correct definition. In most cases we drop the subscript x denote the distribution function as F x. We define the densit function of the rom variable, x, as M. Budge, Jr Jan

5 df x fx x f x. (3-9) dx We can relate the densit function back to the distribution function, probabilit, b x x P x F x f d. (3-10) The probabilit P x x x 1 x 1 x x1 is given b P x x f d. (3-11) For discrete rom variables, f x consists of a series of impulse functions (Dirac delta functions) located at the values that the rom variables can take on. The weight on each impulse is equal to the probabilit associated with the rom variable. B this we mean that the weight is the probabilit associated with the event containing all outcomes for which x xk. In equation form k x k k f x P x x x. (3-1) As an example, the densit function associated with the die experiment is (3-13) k1 f x x k The densit function of a continuous rom variable is a continuous function of x. Rom variables are usuall referenced b the densit function associated with them. Thus a Gaussian rom variable is a rom variable whose densit function is the Gaussian function, 1 x f x e. (3-14) We can derive two ver important properties of the densit function from Equations Specificall: f x x 0 f x dx Mean, Variance Moments of Rom Variables Rather than work directl with rom variables their densit functions we often work with functions of rom variables. We do this for M. Budge, Jr Jan

6 much the same reason we transitioned from set theor probabilities to rom variables densit functions our abilit (or inabilit) to perform the mathematics. In particular, we don t have the mathematical tools to work directl with rom variables /or densit functions. Actuall, we do have the tools but, as ou will recall from rom variable theor, the are primitive difficult to use. The particular functions that we are interested in are termed the various moments of the rom variable, x. The first of these is the first moment of x, which is also called the mean. It is defined as E x. (3-15) xf x dx With this, the reader can show that a Gaussian rom variable with the densit function in Equation 3-1 has a mean of. Two important moments of x are its second moment its second central moment. The first is termed the mean squared value of x the second is termed the variance of x. The square root of the variance is termed the stard deviation of x. The mean squared value variance are given b E x x f x dx (3-16) x E x f x dx, (3-17) respectivel Multiple Rom Variables For some experiments we can define several related or unrelated rom variables. For example, if two dice are used in the experiment, separate rom variables, x, can be assigned to each die. For the case of two rom variables we define joint distribution joint densit functions as, x F x P x (3-18) f x, respectivel. F x,, (3-19) x An example of the densit function for two, jointl Gaussian rom variables is f 1 1 x r x x x x, exp 1 x x x 1r r, (3-0) M. Budge, Jr Jan 018 0

7 's where the definitions of the 's should be obvious r is the correlation coefficient defined b x C x, r (3-1) C x, below. is the covariance between the rom variables x is defined If two rom variables, x, are independent we can write their joint densit as, f x f x f. (3-) The reverse also holds. If Equation 3- is valid then x are independent. We can relate the marginal densities of x to their joint densities b the equations, f x f x d (3-3), f f x dx. (3-4) As with single rom variables, we use joint moments of joint rom variables in lieu of using the actual rom variables, marginal densities joint densities. Again, the reason for this is our depth of mathematical tools. The two specific moments we use for two rom variables are their first joint moment their first, joint, central moment. The former we term the correlation the latter we term their covariance. The correlation is given b, x, R x E xf x dxd (3-5) the covariance is given b C x, E x x x x f x, dxd. (3-6) Some properties terms associated with correlation covariance are as follows. If Rx, 0 then x are orthogonal. If C x, 0 then x are uncorrelated. If x /or are zero mean then Rx, C x,. If x are independent, the are also uncorrelated. The reverse is not true. M. Budge, Jr Jan 018 1

8 If x are uncorrelated jointl Gaussian, the are also independent. 3.4 Rom Processes Introduction In the previous section we discussed the concept of rom variables wherein we assigned a number to experiment outcomes. In some experiments it is more appropriate to assign time functions to experiment outcomes. Examples of this might be where the experiment outcome is the selection of a generator or observation of the voltage at the output of a receiver. In the first case, we might assign sinusoidal time functions (see Figure 3-1) to each experiment outcome. The sinusoids might be of a different amplitude, frequenc relative phase for each experiment outcome (generator selection). For the second case we might assign time functions such as those of Figure 3-. Figure 3-1. Two samples of a rom process generator outputs M. Budge, Jr Jan 018

9 Figure 3-. Two samples of rom processes receiver outputs If we assign a time function,,, to each experiment outcome,, the collection of time functions is termed a rom process or a stochastic process. As we did with rom variables, we will not explicitl denote the dependenc on, as. write We note that, can represent four things 1. A famil of time functions for t variable. This is a stochastic process.. A single time function for t variable fixed. 3. A rom variable for t fixed variable. 4. A number for t fixed. We use representations 3 to help us devise was to deal with rom processes. Specificall we will use rom variable theor time-function, is a rom variable for a specific analsis techniques. Indeed, since value of t we can use the densit functions to characterize it. Thus, for a time t,, is the distribution function of, x, F P t x (3-7) its densit is f x, t F x, t x (3-8) We include the variable t in the expressions for the distribution densit functions to denote the fact that the characteristics (i.e., densit function) of the M. Budge, Jr Jan 018 3

10 rom variable, can change with time. F x, t f, the first-order distribution densit of the rom process. are termed If we consider two instants of time, t a t b, we can describe the characteristics of the rom variables at those times in terms of their firstorder densities the joint densit of the rom variables at the two times; f x, x, t, t. The joint densit is termed the second-order densit specificall of the rom process Autocorrelation, Autocovariance, Crosscorrelation, Crosscovariance As with rom variables, we find it easier to use the moments of the rom process rather than the rom process or its various densities. To this end, we define the mean of a rom process as x, t E t xf dx. (3-9) Note that the mean is, in general, a function of time. We define the autocorrelation autocovariance of a rom process as, x x R t t E t t (3-30) C t, t E t t (3-31) a b a a b b respectivel. The autocorrelation of a rom process is equivalent to the correlation between two rom variables the autocovariance of a rom process is equivalent to the covariance between two rom variables. Note that the autocorrelation autocovariance describe the relation between two rom variables that result b considering samples of a single rom process at two times. The fact that we are considering two rom variables leads to the use of correlation covariance. We use the prefix auto to denote the fact that the rom variables are derived from a single rom process. If we have two rom processes, crosscorrelation crosscovariance as t, we define their Rx ta, tb E a tb (3-3) C t, t E t t t (3-33) x a b a x a b b respectivel. We use the prefix cross to denote that the two rom variables at the two times are now derived from two, different rom processes. At this point we want to briefl delineate some properties of rom processes, autocorrelations, aurocovariances, crosscorrelations crosscovariances. M. Budge, Jr Jan 018 4

11 First we note that if is zero mean then, Also, if. If t a t b If /or t C t, t R t, t t, t. a b C t, t R t, t t, t are zero mean then then C t, t t, the variance of a a a x a b x (evaluated at Cx ta, tb 0 ta, tb we sa that the rom processes are uncorrelated. t t a t If the crosscorrelation of two rom processes is zero for all times then the rom processes are orthogonal. C t, t 0 t t we sa that the rom process If will discuss this again shortl). If the rom variables then the processes, x t a t t b are independent for all are independent. is white (we t a If two processes are independent the are also uncorrelated. The reverse is not normall true. However, if the processes are jointl Gaussian uncorrelated, the are also independent. Two processes are jointl Gaussian if all orders (first, second, etc.) of their individual densit functions are Gaussian functions if all orders of their joint densit functions are Gaussian. We sa that a stochastic process,, is strict sense stationar if the statistics of of this are that the first order densit of order densit functions of in the above sentence can be written in equation form as: f x, t f x, f x, x, t, t f x, x, t t, f x, x, x, t, t, t f x, x, x, t t, t t are independent of a shift in the time origin. The implications is independent of t. all higher depend onl upon relative time. The statements a b c a b c a b c a b a c so forth. As a result of the above we can write the first second moments as x, t E t xf dx xf x dx, (3-34), R t t R (3-35), C t t C. (3-36) We sa that two rom processes, ). t b t are jointl strict sense stationar if each is strict sense stationar if their joint statistics are independent of a shift in the time origin. If two rom processes are jointl strict sense stationar we can write M. Budge, Jr Jan 018 5

12 x, R t t R (3-37) x x, C t t C. (3-38) x The requirement for strict sense stationarit is usuall more restrictive than necessar. Thus, we define a weaker condition b saing that a rom process is wide sense stationar if Equations 3-34, are valid for that rom process. We sa that two processes, sense stationar if the are each wide sense stationar Equations are valid for the two rom processes. t, are jointl wide As a final definition, we define white noise or a white rom process. A rom process,, is white if C t, t E t t t (3-39) where is the Dirac delta function. The above discussions have considered continuous-time sstems. The concepts in the previous discussions can be extended to discrete-time sstems b replacing the time variables b multiples of some sample period, T. That is, t kt, t a k a T, t kt mt, so forth where k, m are integers. When we stud sampled data sstems we often drop the T variable simpl use the integers as arguments of the various variables. With this we can write the various means, autocorrelations, etc. of Equations 3-8 through 3-38 as x, k E k xf x k dx, (3-40), x x R k k E k k, (3-41) C k, k E x k k x k k, (3-4) a b a a b b Rx k, k E x k k, (3-43) C k, k E x k k k k (3-44) x a b a x a b b similarl for the remainder. For white, discrete time rom processes we write C k m, k E x k m k m x k k k s m (3-45) where m s m is the Kronecker delta function is defined as s 1 if m 0. (3-46) 0 if m 0 k a 3.5 Vector Rom Processes M. Budge, Jr Jan 018 6

13 When we work with Kalman filters we will generall work with state variables. As such, we will be concerned with vector rom processes. That is, collections of rom processes that we represent using vectors. Because of this we need to define some vector forms of the mean, autocorrelation, autocovariance, etc. equations that we defined in the previous pages. To start we represent the mean of a continuous time, vector rom process as t t 1 E x1 t E x t t Ex t. (3-47) n t Exn t R In a similar fashion we can write T t, t E xt x t x1 a x1 b x1 a x b x1 a xn b x a x1 b x a x b x a xn b E t t E t t E t t E t t E t t E t t E x1t E xt E n a b n a b n a n b a, b a, b, n a b,,, Rx 1 t R 1 x1 t R x1x t t R t t R t t R t t 1 n R t t R t t R t t n 1 n n n x x a b x x a b x x a b,,, x x a b x x a b x x a b. (3-48) The T superscript notation in Equation 3-48 denotes the transpose operation. It will be noted that the last matrix of Equation 3-85 contains both autocorrelation crosscorrelation terms. As a result, we term this matrihe autocorrelation matrix. We will often need to discuss the crosscorrelation between two, different, vector rom processes. This is again a matrix, which we will term the crosscorrelation matrix. This matrix has the form T t, t E t t R x x x1 a 1 b x1 a b x1 a m b x a 1 b x a b x a m b E t t E t t E t t E t t E t t E t t E 1t E t E t n a b n a b n a m b M. Budge, Jr Jan 018 7

14 a, b a, b, m a b,,, Rx 1 t t R 1 x1 t t R x1 t t R t t R t t R t t 1 m R t t R t t R t t n 1 n n m x a b x a b x a b,,, x a b x a b x a b (3-49) It will be noted that this equation contains onl crosscorrelation terms. It will also be noted that, while the autocorrelation matrix is alwas square, the crosscorrelation matrix is an n m matrix is most often not square. We define the autocovariance matrix b the equation x, x C t t E t t t t. (3-50) a b a a b b If we let t a t b t C, x we obtain the covariance matrix T x P T t t E t t t t t. (3-51) Finall, we write the crosscovariance matrix as, x C t t E t t t t (3-5) x a b a x a b b We can define the sampled data versions of Equations 3-47 through 3-5 b replacing the time variable, t, with the stage variable, k. Some notes on vector rom processes Two vector rom processes are uncorrelated if, onl if, C t, t 0 t, t. x T A continuous-time, vector rom process is white if, onl if, C t, t P t. A discrete-time, vector rom process is white if, onl if, C k m k P k m., s A vector rom process is Gaussian if all the elements of the vector are jointl Gaussian. M. Budge, Jr Jan 018 8

15 3.6 Problems 1. Derive the relations given b Equations Use Equations to show that f x 0 x f xdx 1 3. Show that the Gaussian densit of Equation 3-1 has a mean of. 4. Show that Rx, C x, if x 0 /or Show that two independent rom variables are also uncorrelated. 6. Show that two uncorrelated, jointl Gaussian rom variables are also independent. 7. Show that two independent rom processes are also uncorrelated. 8. Show that two uncorrelated, jointl Gaussian rom processes are also independent. T 9. Show that R k, k R k, k x a b x b a. 10. Show that P k is a non-negative definite matrix. M. Budge, Jr Jan 018 9